Proceed with Caution, you are entering dangerous territory.
Expanding inspiration in this manner will make it more powerful (by definition). Adding dramatically more options to any ability will do this.
There are also some soft differences between granting success versus granting failure. All party members succeeding once will have a very different feel than a monster who fails four to six consecutive checks.
But these aren't the biggest threat you're going to face. That one belongs to the pure casters...
Save or Suck
While an attack roll on a spell versus a saving throw for its target often seems like an arbitrary distinction, they aren't always. There is a category of spells often referred to as "Save or Suck," which almost universally favor saving throws instead of attack rolls. Spells like Hold Person, Banishment, Feeblemind, Entangle, and Polymorph all inflict near-catastrophic effects on a failed saving throw.
Under your proposed house rule, an optimal tactic would be for players to transfer their inspiration to the full casters in the group (Wizard, Cleric, Druid, Sorcerer), who then uses it to cast repeated save or suck spells on any monster who stands above the crowd.
This has two very negative consequences:
Players are now under social pressure to transfer their inspiration elsewhere, rather than using it to help their own characters shine.
"Boss" fights become incredibly difficult to stage, as any boss has to deal with multiple saves against incapacitating effects with disadvantage applied.
This is not a fatal flaw...
This is not a fatal flaw to the house rule. Monsters written to be boss monsters typically have auto-save abilities, because even without disadvantage save or suck spells are pretty devastating.
In addition, exploiting it requires players to play at a somewhat optimized level. I would not be surprised for an individual group to either not see this tactic, or to be chivalrous about not exploiting it.
But be careful. Make sure that everyone involved is aware that you may back this change out early if it causes problems.
The GM must reveal the number on the die
Whenever a player has an ability to do something after knowing "the roll, but not the result", the GM must either tell or show the player the number on the dice, as confirmed in this pair of unofficial tweets from June 2016 by rules designer Jeremy Crawford:
Bard Combat Inspiration use with reaction to add the roll to AC "AFTER SEEING THE ROLL" So DM must show his roll to player?
If a player character ability relies on knowing a monster's roll, a DM may say what the roll is or show it
Master @JeremyECrawford, you intended that DM must communicate only the number on the dice (without added any modifiers), right?
That's correct.
Relevant modifiers or target numbers do not have to be revealed, only the raw rolls of the dice.
Therefore, it is courteous and convenient for the GM to roll the dice openly whenever there is a chance for a player to use their inspiration, ask whether the player wants to use their inspiration, and declare the hit/miss after the player replies.
Best Answer
From the Player's Basic Rules v 0.2, page 35:
Generally, if something lets you wait until after the roll and see how you did before you use it, it specifies that. "When you make" a roll suggests that you use it at the same time as the roll, which isn't that helpful, but its effect is to give you advantage, which says (page 57):
This is more helpful, because it says you roll the second d20 when you make the roll. So advantage doesn't allow you to make the roll, then use your second d20; you roll them both at the same time. Since inspiration just gives advantage, it seems clear that you have to decide to use it before you roll.